Extensive Condensation in a model of Preferential Attachment with Fitnesses

Freeman, Nic; Jordan, Jonathan

doi:10.48550/arxiv.1812.06946

Cited by 5 publications

(14 citation statements)

References 17 publications

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“…The new vertex forms an edge between itself and the vertex with the highest fitness of the r selections. It is shown in [8] that again condensation can occur in this model.…”

Section: Introductionmentioning

confidence: 70%

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Condensation in preferential attachment models with location‐based choice

Haslegrave

Jordan

Yarrow

2019

Random Struct Algorithms

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We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting r vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant α. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in α below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.

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“…The new vertex forms an edge between itself and the vertex with the highest fitness of the r selections. It is shown in [8] that again condensation can occur in this model.…”

Section: Introductionmentioning

confidence: 70%

“…The choice model is combined with fitness in the model studied by Freeman and Jordan [8] in which each vertex has its own fitness value; the new vertex joins the graph G n by using preferential attachment to select r vertices from G n . The new vertex forms an edge between itself and the vertex with the highest fitness of the r selections.…”

Section: Introductionmentioning

confidence: 99%

Condensation in preferential attachment models with location‐based choice

Haslegrave

Jordan

Yarrow

2019

Random Struct Algorithms

Self Cite

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“…By viewing the CTBP as a reinforced P ólya's urn, it is also possible to study condensation by establishing the strict positivity in the limit of the cumulative degree for vertices with high fitness. This is in fact the first approach to the mathematical study of condensation, pioneered by Borgs et al 2007, see also Freeman and Jordan 2018.…”

Section: Proofs Of the Resultsmentioning

confidence: 97%

Dynamical fitness models: evidence of universality classes for preferential attachment graphs

Cipriani¹,

Fontanari²

2019

Preprint

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In this paper we define a family of preferential attachment models for random graphs with fitness in the following way: independently for each node, at each time step a random fitness is drawn according to the position of a moving average process with positive increments. We will define two regimes in which our graph reproduces some features of two wellknown preferential attachment models: the Bianconi-Barabási and the Barabási-Albert models. We will discuss a few conjectures on these models, including the convergence of the degree sequence and the appearance of Bose-Einstein condensation in the network when the drift of the fitness process has order comparable to the graph size.

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“…[22,23] consider a version where the attachment function can be sublinear in the degree. In [18,20,24,28] vertices are equipped with a fitness and in [38] the arriving vertices have a power of choice. Spatial variants where vertices have a location in an underlying Euclidean space are studied in [2,35,36].…”

Section: Introductionmentioning

confidence: 99%

“…We have initiated work in this direction and hope to communicate results on it soon. Some frequently studied global properties are the size of the giant component and its robustness against site or edge percolation [23,26,30,36], and condensation phenomena [10,18,20,28,38]. 1.3.…”

Section: Introductionmentioning

confidence: 99%

Distance evolutions in growing preferential attachment graphs

Jorritsma¹,

Komjáthy²

2020

Preprint

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We study the evolution of the graph distance and weighted distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with power-law exponent τ ∈ (2, 3), sample two vertices ut, vt uniformly at random when the graph has t vertices, and study the evolution of the graph distance between these two fixed vertices as the surrounding graph grows. This yields a discrete-time stochastic process in t ≥ t, called the distance evolution. We show that there is a tight strip around the function 4 log log(t)−log(1∨log(t /t))∨ 4 that the distance evolution never leaves with high probability as t tends to infinity. We extend our results to weighted distances, where every edge is equipped with an i.i.d. copy of a non-negative random variable L.

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Extensive Condensation in a model of Preferential Attachment with Fitnesses

Cited by 5 publications

References 17 publications

Condensation in preferential attachment models with location‐based choice

Condensation in preferential attachment models with location‐based choice

Dynamical fitness models: evidence of universality classes for preferential attachment graphs

Distance evolutions in growing preferential attachment graphs

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